The role of the spectrum of the laplace operator on S-2 in the H-bubble problem

Let H be a given smooth scalar function on R^3. An H-bubble is an S^2-type parametric surface in R^3 having mean curvature H(p) at every regular point. In the present paper we look for (H(|\cdot|)+\epsilon k)-bubbles, where H is a radial prescribed curvature satisfying H(1)=1, k is given and \epsilon is a small parameter.